Trajectory planning in parallel kinematic manipulators using a constrained multi-objective evolutionary algorithm

Generating manipulator trajectories considering multiple objectives with kinematics and dynamics constraints is a non-trivial optimization. In this paper, a constrained multi-objective genetic algorithm (MOGA) based technique is proposed to address this problem for a general motor-driven parallel kinematic manipulator. The planning process is composed of searching for a motion ensuring the accomplishment of the assigned task, minimizing the traverse time, and expended energy subject to various constraints imposed by the associated kinematics and dynamics of the manipulator. This problem is treated via an adequate parametric path representation in the task space of the moving platform, and then the use of the constrained MOGA for solving the resulted nonlinear multi-objective optimization problem. Simulation results are presented for the trajectories of the parallel kinematic manipulator, and a subsequent comparison with the weighted sum method is also carried out.     

Kinematic modeling and dexterity evaluation of a PS-RRS-2RUS parallel manipulator used for controllable pitch propeller

This article investigates the kinematic modeling and dexterity evaluation of a PS-RRS-2RUS parallel manipulator. The design concept and mobility analysis of the manipulator are first addressed utilizing the screw theory. Second, the decoupled and closed-form kinematic solutions with multiple configurations are solved through vector method. Then, the analytical expressions for the velocity relationships are derived into a closed-form Jacobian matrix, which is then used to distinguish the singular postures. Finally, the workspace with fixed orientation is calculated and its dexterity is evaluated. The numerical simulations and validation are conducted in a case study.     

Complete kinematics of a serial–parallel manipulator formed by two Tricept parallel manipulators connected in serials

Complete kinematic is an essential and a challenging work for series–parallel manipulators (S–PMs). This paper studied the complete kinematic of a 2(3-SPS+UP) series–parallel manipulator. First, a S–PM formed by two well-known Tricept parallel manipulators (PMs) connected in serial is presented. Second, the forward and inverse displacements are studied using sylvester dialytic elimination method. Third, the forward and inverse Jacobian matrices are established based on integrating the constraint and coupling information of the single PMs into the S–PM. Fourth, simple and compact formulae for the forward and inverse acceleration are derived using vector approach. Finally, the workspace of this S–PM is constructed using CAD variation geometry approach. The results show that the 2(3-SPS+UP) S-PM has multiple forward and inverse position solutions. The existence and uniqueness of the forward, inverse Jacobian matrices and the acceleration formula are shown from their explicit form. The workspace analysis shows that this S–PM has large workspace. The research works provided a theoretical basis for the novel 2(3-SPS+UP) S–PM, as well as a feasible approach for establishing the complete kinematics for S–PMs.     

Inverse dynamics analysis of a general spherical star-triangle parallel manipulator using principle of virtual work

Inverse dynamics of a general model of a spherical star-triangle (SST) parallel manipulator (Enferadi and Akbarzadeh Tootoonchi, Robotica 27:663–676, 2009) is the subject of this paper. This manipulator is of type 3-RRP, has good accuracy and relatively a large workspace which is free of singularities (Enferadi and Akbarzadeh Tootoonchi, Robotica, Revised paper, 2009). First, inverse kinematics utilizing the angle axis representation is solved. Next, velocity and acceleration analysis as well as link Jacobian matrices are obtained in invariant form. Finally, a systematic approach based on the principle of virtual work and the concept of link Jacobian matrices is presented. This method allows elimination of constraint forces and moments at the passive joints from motion equations. It is shown that the dynamics of the manipulator can be reduced to solving a system of three linear equations with three unknowns. Moreover, a computational algorithm for solving the inverse dynamics is developed. Two examples with different trajectories for the moving spherical platform are presented and motor torques are obtained. Results are verified using a commercial dynamics modeling package.     

Forward position analysis of 6-3 Linapod parallel manipulators

In this study closed-form solutions to the forward kinematic problems are obtained for a particular type of six degree-of-freedom parallel manipulator called 6-3 Linapod. The 6-3 Linapod parallel manipulators have a 6-3 PSS (or PUS) structure, and forward kinematic solutions are obtained by using the solution procedure for 6-3 SPS (or UPS) manipulators. In this procedure, a 6-3 Linapod is first transformed into its equivalent mechanism, namely an inclined 3RS manipulator, and then the condition that the three spherical joints on the moving platform form an equilateral triangle leads us to obtain three polynomial equations in three unknowns. These equations are solved by using Sylvester dialytic elimination method. Each set of real roots corresponds to a particular configuration of the manipulator. Solutions so obtained are verified by performing inverse position analysis. A method to identify configurations containing crossed links is presented in this study, which is based on the interpretation of link crossing as intersection of a link with a triangle, whose vertices are positions of joints on the corresponding links.     

Inverse dynamics of the HALF parallel manipulator with revolute actuators

Recursive matrix relations for kinematics and dynamics of the HALF parallel manipulator are presented in this paper. The prototype of this robot is a spatial mechanism with revolute actuators, which has two translation degrees of freedom and one rotation degree of freedom. The parallel manipulator consists of a base plate, a movable platform and a system of three connecting legs, having wide application in the fields of industrial robots, simulators, parallel machine tools and any other manipulating devices where high mobility is required. Supposing that the position and the motion of the moving platform are known, an inverse dynamics problem is solved using the principle of virtual powers. Finally, some iterative matrix relations and graphs of the torques and powers for all actuators are analysed and determined. It is shown that this approach is an effective means for kinematics and dynamics modelling of parallel mechanisms.     

Par2: a spatial mechanism for fast planar two-degree-of-freedom pick-and-place applications

This paper introduces a new two-degree-of-freedom (dof) parallel manipulator producing two translations in the vertical plane. One drawback of existing robots built to realize these dof is their lack of transversal stiffness, another one being their limited ability to provide very high acceleration. Indeed, these architectures cannot be lightweight and stiff at the same time. The proposed parallel architecture is a spatial mechanism which guarantees a good transversal stiffness. It is composed by two actuated kinematic chains, and two passive chains built in the transversal plane. The key feature of this robot comes from the two passive chains which are coupled to create a kinematic constraint: the platform stays in one plane. A stiffness analysis shows that the robot can be lighter and stiffer than a classical 2-dof mechanism. A prototype of this robot is presented and preliminary tests show that accelerations above 400 ms⁻¹ can be achieved while keeping a low tracking error.     

Analysis of a 6-DOF redundantly actuated 4-legged parallel mechanism

In this paper, we propose a new 4-legged 6-DOF redundantly actuated UPS parallel mechanism. Kinematics, singularity and dynamics analyses of the proposed mechanism are addressed. Inverse kinematics has a closed-form solution, whereas the forward kinematics has a semi-closed form solution. Also, the Jacobian matrix has been determined using the concept of reciprocal screws. Using the principle of virtual work and minimum norm method, a general formulation for the inverse dynamics of redundantly actuated parallel manipulator is presented. Moreover, the proposed redundant mechanism has been compared with a similar but nonredundant mechanism in three aspects: reachable points, and actuation forces under static and dynamic conditions. We show that the redundancy decreases singular points, and dramatically reduces the actuators’ forces and torques.     

Kinematic analysis of a 5-R SP parallel mechanism with?centralized motion

The paper presents a kinematic analysis of a parallel mechanism, referred to here as a mechanism with centralized motion. The paper includes a proof, based on the geometry of the mechanism, that the platform exhibits centralized motion. An interesting feature of this parallel mechanism is that it is partially collapsible which may be beneficial in practical applications where storage space is limited. The platform is connected to a base, regarded as fixed in this paper, by five identical legs where each leg is a three-link chain connected by a revolute joint, a spherical joint, and a prismatic joint. The result is that the platform has a screw motion about an axis which is perpendicular to the base and passes through the centroids of the base and the platform, for all positions of the platform. The pitch of the instantaneous screw depends on the platform assembly configuration and is a function of the platform position and orientation. To complete the kinematic study, the paper includes closed-form solutions to the inverse and forward position and velocity problems. Finally, the paper includes several numerical examples to illustrate some of the key features of this novel parallel mechanism.     

Prototype design of a translating parallel robot

The paper describes the mechanical design of a parallel manipulator for motions of pure translation, whose kinematic analysis has shown very good performances such as a large workspace and small overall dimensions of the mobile platform; in particular, the “Cartesian” structure of the machine allowed to obtain constant accuracy and kinematic properties throughout the workspace. The structural design has minimized the mass of the moving links and, by the combined use of FEM and multibody codes, allowed to take into consideration the high stresses coming from inertial forces and to evaluate a-priori the resulting dynamic properties. A physical prototype has just been built in order to validate the developed models and assess the actual robot performances in real operating conditions. The present research has been partially co-funded by the Italian Ministry of Research and University and by the Polytechnic University of Marche under PRIN03 project “Design and prototyping of application-oriented mini-PKM”.     

Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations

The paper presents a new structural synthesis approach of fully-isotropic translational parallel robotic manipulators (TPMs) based on the theory of linear transformations. A TPM is a 3-DOF (degree of freedom) parallel mechanism whose output link, called platform, can achieve three independent orthogonal translational motions with respect to the fixed base. The manipulators presented in this paper have three legs connecting the moving platform and the base (fixed platform). Only one kinematic pair per leg is actuated by a linear motor situated on the fixed base. A one-to-one correspondence exists between the actuated joint space and the operational space of the moving platform. The Jacobian matrix of fully-isotropic TPMs presented in this paper is the identity 3×3 diagonal matrix throughout the entire workspace. The synthesis method proposed in this paper allows us to obtain all structural solutions of fully-isotropic TPMs in a systematic manner. Overconstrained/isostatic solutions with elementary/complex and identical/different legs are obtained. Fully-isotropic TPMs have the advantage of simple command and important energy-saving due to the fact that, for a unidirectional translation, only one motor works as in a serial translational manipulator.     

Inverse dynamics of a 3-PRC parallel kinematic machine

Recursive matrix relations for kinematics and dynamics analysis of a three-prismatic-revolute-cylindrical (3-PRC) parallel kinematic machine (PKM) are performed in this paper. Knowing the translational motion of the platform, we develop first the inverse kinematical problem and determine the positions, velocities and accelerations of the robot’s elements. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work and the results in the framework of the Lagrange equations with their multipliers can be verified. Finally, compact matrix equations and graphs of simulation for input force and power of each of three actuators are obtained. The investigation of the dynamics of this parallel mechanism is made mainly to solve successfully the control of the motion of such robotic system.     

Dynamics analysis of a parallel hip joint simulator with four degree of freedoms (3R1T)

For a hip joint simulator with a 3SPS+1PS spatial parallel manipulator as the core module, a formulation based on the Kane equation was derived for the dynamic characteristics of the simulator from the kinematics analysis of the model. The relationships of the velocities and accelerations between the moving platform and active branched-chains were deduced. The velocity and angular velocity components of the moving platform were served as the generalized velocities. And the dynamic model was established by obtaining the generalized active forces and inertial forces. Then the driving forces and powers of the active branched-chains and the constraint reaction forces of the intermediate branched-chain were simulated in the numerical method. The results showed that the active driving forces of the branched-chains reached their respective maximum when the moving platform rotated into 0.13^° around X-axis, 2^° around Y-axis, and 18^° around Z-axis. And the intermediate branched-chain needed to balance the driving and inertia forces, as well as support the moving platform and load the force of hydraulic cylinder. Therefore, the maximum constraint reaction force of the intermediate branched-chain is along the Z-axis. The research works provided a theoretical basis for the design of the active branched-chains driving system and the structural parameters of the intermediate branched-chain, as well as for the control system.     

Design and analysis of a compliant parallel pan-tilt platform

In combination of the advantages of both parallel mechanisms and compliant mechanisms, a compliant parallel mechanism with two rotational degrees of freedom is designed to meet the requirement of a lightweight and compact pan-tilt platform. Firstly, two commonly-used design methods i.e. direct substitution and Freedom and Constraint Topology are applied to design the configuration of the pan-tilt system, and similarities and differences of the two design alternatives are compared. Then inverse kinematic analysis of the candidate mechanism is implemented by using the pseudo-rigid-body model, and the Jacobian related to its differential kinematics is further derived to help designer realize dynamic analysis of the 8R compliant mechanism. In addition, the mechanism’s maximum stress existing within its workspace is tested by finite element analysis. Finally, a method to determine joint damping of the flexure hinge is presented, which aims at exploring the effect of joint damping on actuator selection and real-time control. To the authors’ knowledge, almost no existing literature concerns with this issue.     

Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics

The problem of reducing the body-attached coordinate system to the reference (programmed) coordinate system moving relative to the fixed coordinate system with a given instantaneous velocity screw along a given trajectory is considered in the kinematic statement. The biquaternion kinematic equations of motion of a rigid body in normalized and unnormalized finite displacement biquaternions are used as the mathematical model of motion, and the dual orthogonal projections of the instantaneous velocity screw of the body motion onto the body coordinate axes are used as the control. Various types of correction (stabilization), which are biquaternion analogs of position and integral corrections, are proposed. It is shown that the linear (obtained without linearization) and stationary biquaternion error equations that are invariant under any chosen programmed motion of the reference coordinate system can be obtained for the proposed types of correction and the use of unnormalized finite displacement biquaternions and four-dimensional dual controls allows one to construct globally regular control laws. The general solution of the error equation is constructed, and conditions for asymptotic stability of the programmed motion are obtained. The constructed theory of kinematic control of motion is used to solve inverse problems of robot-manipulator kinematics. The control problem under study is a generalization of the kinematic problem [1, 2] of reducing the body-attached coordinate system to the reference coordinate system rotating at a given (programmed) absolute angular velocity, and the presentedmethod for solving inverse problems of robotmanipulator kinematics is a development of the method proposed in [3–5].     

A new method for optimum design of parallel manipulator based on kinematics and dynamics

This paper presents a new method for the optimum design of parallel manipulators by taking both the kinematics and dynamic characteristics into account. The optimum design of a 3-DOF 4-RRR planar parallel manipulator with actuation redundancy is investigated to demonstrate the method. The kinematic performance indices such as the conditioning index, the velocity index, and workspace area are analyzed. Further, the dynamic dexterity, which is used to evaluate the dynamic characteristics, is investigated. The corresponding atlases are represented graphically in the established design space. Based on these atlases, the geometrical parameters without dimension are determined. Then the optimum dimensional parameters are achieved based on the optimum non-dimensional result. By using the method proposed in this paper, the designer can obtain the optimum result with respect to both kinematic performance indices and dynamic performance indices. Since the dynamic performance is considered in the process of optimum design by using the method proposed in this paper, it is expected to realize the high dynamics of parallel manipulators.     

Dynamic system of an engine with spatially rocking links: a mathematical model

A mathematical model of a diesel with swashplates is proposed. The results of studying its design, kinematics, and dynamics are presented. The kinematic chain of the statically determinate mechanism of the engine is schematized. The kinematic relations are obtained taking into account the presence of Hooke’s joints and swashplates. A mathematical model of the dynamic system of the engine is described. A numerical example is given. The kinematics and dynamic processes of the engine are studied     

Comparison of 3-RPR planar parallel manipulators with?regard to their kinetostatic performance and sensitivity to geometric uncertainties

This paper deals with the sensitivity analysis of 3-RPR planar parallel manipulators. First, the manipulators under study as well as their degeneracy conditions are presented. Then, an optimization problem is formulated in order to obtain their maximal regular dexterous workspace. Moreover, the sensitivity coefficients of the pose of the manipulator moving platform to variations in the geometric parameters and in the actuated variables are expressed algebraically. Two aggregate sensitivity indices are determined, one related to the orientation of the manipulator moving platform and another one related to its position. Then, we compare two non-degenerate and two degenerate 3-RPR planar parallel manipulators with regard to their dexterity, workspace size and sensitivity. Finally, two actuating modes are compared with regard to their sensitivity.     

Design and analysis of a novel 6-DOF redundant actuated parallel robot with compliant hinges for high precision positioning

This paper presents the design and modeling of a new 6-DOF 8-PSS/SPS compliant dual redundant parallel robot with wide-range flexure hinges. This robot can achieve either high accurate positioning or rough positioning as well as a 6-DOF active vibration isolation and excitation to the payload placed on the moving platform. Adopting a kind of wide-range flexure hinge, we establish the kinematics model of the macro parallel mechanism system via the stiffness model and Newton–Raphson method, then we build up the dynamics model using Kane’s method for the micro-motion system. The investigations of this paper will provide suggestions to improve the structure and control algorithm optimization for a novel compliant dual redundant parallel mechanism in order to achieve the feature of larger workspace, higher motion precision and better dynamic characteristics. The results will be helpful in modifying the structure of the prototype platform to enhance its high kinematics and dynamics properties.     

Dynamics and control of variable geometry truss manipulator

Variable geometry truss manipulator(VGTM) has potential to work in the future space applications, of which a dynamic model is important to dynamic analysis and control of the system. In this paper, an approach is presented to model the dynamic equations of a VGTM by independent variables, which consists of two double-octahedral truss units and a 3-revolute-prismatic-spherical(3-RPS) parallel manipulator. In this approach, the kinematic recursive relations of two adjacent bodies and geometric constrains are used to deduce the kinematic equations of the VGTM, and Jourdain's velocity variation principle is adopted to establish the dynamic equations of the system. The validity of the proposed dynamic model is verified by comparison of numerical simulations with the software ADAMS. Besides, an active controller for trajectory tracking of the system is designed by the computed torque method. The effectiveness of the controller is numerically proved.